## The Annals of Statistics

### Convergence of algorithms for reconstructing convex bodies and directional measures

#### Abstract

We investigate algorithms for reconstructing a convex body K in ℝn from noisy measurements of its support function or its brightness function in k directions u1,…,uk. The key idea of these algorithms is to construct a convex polytope Pk whose support function (or brightness function) best approximates the given measurements in the directions u1,…,uk (in the least squares sense). The measurement errors are assumed to be stochastically independent and Gaussian.

It is shown that this procedure is (strongly) consistent, meaning that, almost surely, Pk tends to K in the Hausdorff metric as k→∞. Here some mild assumptions on the sequence (ui) of directions are needed. Using results from the theory of empirical processes, estimates of rates of convergence are derived, which are first obtained in the L2 metric and then transferred to the Hausdorff metric. Along the way, a new estimate is obtained for the metric entropy of the class of origin-symmetric zonoids contained in the unit ball.

Similar results are obtained for the convergence of an algorithm that reconstructs an approximating measure to the directional measure of a stationary fiber process from noisy measurements of its rose of intersections in k directions u1,…,uk. Here the Dudley and Prohorov metrics are used. The methods are linked to those employed for the support and brightness function algorithms via the fact that the rose of intersections is the support function of a projection body.

#### Article information

Source
Ann. Statist., Volume 34, Number 3 (2006), 1331-1374.

Dates
First available in Project Euclid: 10 July 2006

https://projecteuclid.org/euclid.aos/1152540751

Digital Object Identifier
doi:10.1214/009053606000000335

Mathematical Reviews number (MathSciNet)
MR2278360

Zentralblatt MATH identifier
1097.52503

#### Citation

Gardner, Richard J.; Kiderlen, Markus; Milanfar, Peyman. Convergence of algorithms for reconstructing convex bodies and directional measures. Ann. Statist. 34 (2006), no. 3, 1331--1374. doi:10.1214/009053606000000335. https://projecteuclid.org/euclid.aos/1152540751

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